Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,6,8,4)(9,10,14,18)(11,15,21,23)(12,16,22,20)(13,17,24,19)(25,26,31,29)(27,35,37,32)(28,38,36,33)(30,34,40,39);(1,3)(2,6)(4,7)(5,8)(9,11)(10,15)(12,17)(13,20)(14,21)(16,24)(18,23)(19,22)(25,27,31,37)(26,32,29,35)(28,30,36,40)(33,34,38,39);(1,2)(3,4)(5,7)(6,8)(9,12)(10,16)(11,19)(13,15)(14,22)(17,21)(18,20)(23,24)(25,26,31,29)(27,32,37,35)(28,33,36,38)(30,34,40,39);(1,4,5,6)(2,3,7,8)(9,12)(10,16)(11,17)(13,23)(14,22)(15,24)(18,20)(19,21)(25,28)(26,33)(27,30)(29,38)(31,36)(32,34)(35,39)(37,40);(1,4)(2,3)(5,6)(7,8)(9,11)(10,15)(12,19)(13,16)(14,21)(17,22)(18,23)(20,24)(25,29,31,26)(27,35,37,32)(28,33,36,38)(30,34,40,39);(1,2)(3,6)(4,8)(5,7)(9,13)(10,17)(11,20)(12,15)(14,24)(16,21)(18,19)(22,23)(25,30)(26,34)(27,36)(28,37)(29,39)(31,40)(32,38)(33,35)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=g,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9121958);